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Unformatted text preview: siti Po Differentiation
y = f(x) k, constant x x2 x3 xn, any constant n ex e kx e
f (x) c ,y) m= Graphs of Common(xFunctions2 x1 x
1 1 y 2 y1
x dy = f'(x) dx 0 1 2x 3x2 nxn1 ex = y ke kx = ky f'(x)e f(x) 1/x Linear
y y = mx + c; m = gradient; c = vertical intercept
y
nt siti Po g ve ie rad (x2, y2) c (x1, y1)
Ne m= y 2 y1 x 2 x1
(x2, y2) Economics Network
Supporting economics in higher education yg e x =
ativ c (x1, y1) y y1 m= 2 x 2 x1
x eg rad i ent 1 1 x ln x y Exponential functions y y 1 m= 2 e 2.7183) is the exponential constant1 x 2 x1 (x1, y1 c
Ne = e x y ati g ve g x y = ex
rad i ent ln kx = log u dv d 1/x d d kx du dv (u( x) v( x )) = e (u( x) v( x )) = dx dx dx dx f'(x)/f(x) dx dx ln f(x) d du dv (u( x) v( x )) = dx dx dx d du dv df d df d (u( x) v((x )) = ( x )) = k k f The sumdifference rule Constant( x )) = k ( k f multiples dx dx dx dx dx dx dx df d du dv d ( k f ( x )) = k (u( x) v( x )) = dx dx dx dx dx d d k df dv dfor k constant du ( k f ( x )) = uv) = u dv + v du ( (uv) = u +v dx dx dx dx dx dx dx dx df d Thed ( k f ( x )) = k product rule The quotient dv + v du rule (uv) = u dx dxdu d dx dv ddx dv x u d dv du v u (uv) = u u + v v dx u dx d d u dx dx dx dx = dx 2 = du 2 dv dx dv v dx v du v d v vu (uv) = u +v d u dx dx d dx The chain ruleu x dv x = dd dx v v u v2 d u dy dy . du dx dx dy dy . du y= If y = (u), where u = u( x ), then = y(u), where u = u( x ), then = If y = du 2 dv dx v dx du dx dx du dx v v u d u dy dy . du dx dx If y = y(u), where u = u( x ), then = = dx v v2 dx du dx dy dy . du If y = y(u), where u = u( x ), then = dx du dx dy dy . du f ( x ) dx f (x , If y = y(u), where u = u( x )) then = dx du dx (x2, y2) 1 1 1 x 1
if a is positive (1) (2) x 1 1 (3) x Graph of y x showing = y=e exponential growth ex Graph of y = showing exponential decay x b / 2a e x Quadratic functions y = ax2 + bx + c
1
if a is positive (1) (2) 1 1 (3) x b / 2a x
(3) Integration TC
b / 2a Maths for Economics
PRINCIPLES AND FORMULAE x if a is negative (2) (1) k, (any) constant c kx + c (1) b2 4ac < 0; (2) b2 4ac = 0;
a x
x2 x n, (n = 1) x 1 = 1/x e e
x kx x2 +c 2 x3 +c 3 x n+1 +c n+1 ln x + c e +c e kx +c k
a
x (3) b2= fixed cost a 4 ac > 0 Total cost functions TC = a + bq TC
if a is negative b / 2a Output (q ) Inverse functions cq2 + dq3 x
(3) (2) (1) y = a/x = ax1 y Example: Unit price elasticity of demand q = a/p = ap1 This leaflet has been produced in conjunction with mathcentre a = fixed cost Output (q ) x www.mathcentre.ac.uk Arithmetic
When multiplying or dividing positive and negative numbers, the sign of the result is given by: + and + gives + and + gives + and gives and gives + e.g. 6 x 3 = 18; e.g. (6) x 3 = 18 e.g. 6 x (3) = 18 e.g. (6) x (3) = 18 21 7 = 3 (21) 7 = 3 21 (7) = 3 (21) (7) = 3 Removing brackets a(b + c) = ab + ac (a + b)2 = a2 + b2 + 2ab; (a + b)(a b) = a2 b2 Algebra
a(b c) = ab ac (a  b)2 = a2 + b2 2ab Sigma Notation
The Greek capital letter sigma is used as an abbreviation for a sum. Suppose we have x values: x1, x2, ... xn and we x1 + n2 . . . xn wish to add them together. The sum n x + x . .. x x1 + 1 2 x. .2 xx .n . x x + n is written x i . . n 1 2 (a + b)(c + d) = ac + ad + bc + bd Formula for solving a quadratic equation Order of calculation First: Second: Third: Fractions Fraction = numerator denominator brackets x and + and If ax2 + bx + c = 0, then x =
Laws of indices b b2 4 ac 2a a m a n = a m+n a0 = 1 am an
1 am = am n ( a m )n = a mn
n a m = a1 / n = a a m / n = n am Adding and subtracting fractions To add or subtract two fractions, first rewrite each fraction so that they have the same denominator. Then, the numerators are added or subtracted as appropriate and the result is divided by the common denominator: e.g.
4 3 16 15 31 + = + = 5 4 20 20 20 3 5 15 Multiplying fractions = 4 3 716 1115 77 31 To multiply two fractions, multiply their numerators and + = + = 5 4 20 20 20 Laws of logarithms y = log b x means b y = x and b is called the base e.g. log 10 2 = 0.3010 means 100.3010 = 2.000 to 4 sig figures Logarithms to base e, denoted loge, or alternatively ln, are called natural logarithms. The letter e stands for the exponential constant, which is approximately 2.7183. x x n .. x + x Note that1x=runs. through all integers (whole numbers) from ii i 1x2i+ x n . . 3 . x 1 1 to n. i =1 for =1 i2 n x i means x1 + x 2 + x 3 So, n instance i 3 x i n i =1 3 x means x + x + x i x 3 x +=1 . . . x i 2 1 i x=1meansmeans125xn +3x 3 x 2 2 2 2 2 1 x2 i x i i x x1 + = + x 2 2+ 3 i i =1 1i means 1 + 2 + 3 + 4 + 5 i 3=1 n Examplex 3means ix 1+ x + x = 5 2 i 2 x i 21 22 23 2 5 =1 means +1++ 3++x4++ 2 + 2 3 2 + x5 x 2 i 5 imeans x i=means 3 1 2 4 2 5 2 1 2 2 2 i ii =1 i 2 means 12 + 2 2 + n 2 + 4 + 5 i1 1 2 = 3 xi x + x + . . . + x i =1 2 n i 5=1 x = i =1 = 1 3 25 2 2 2 i nmeans x + 2 + . 322+ x 2 1 + +x. . x2 i means x 2 usedn+x42 + 52 2 n This notation is 1 12 + 22 + 3n + 4 + 5 calculations. The often in statistical 3 i x = =i x xi =1= i11i n x= + x + . . .x x n x= i means 1 + x = i x1iofnthe 1n i =2x1 +n 2 xn.,.x2, x n and xn is n mean = 1 == 1 quantities, +1 . + ... = i 2 n n n n=1 ( x i x ) x2 n i 5 n x var( x ) x + . . . + x = i =1 i x 2 x1 + =2 2 n2 i n x = 2 means=x 2 2 2 2 +n3 + 4. n + x 2 i i =1 1 n n( x in=1 x i) + x1 + x 2 x i 2 . 5n +. n x 1 i == ==2n=i12n 2 2x 2 i= 1 i ( x i n x ) 2 = var( x ) i =1 )n=1 ix n x i = var( x ) var( x ) = i =1 nx i = x = n i =1 x i x 2 n ( n n The variance is n 2 (x ) =n var(x ) n n x ) + + nx ( xx x sd x . . . + x2 2 x = = =) var( x ) = i i=11 ni i= ( x1i x2 2 i =1 ni n ix 2 2 n in n = n i =1 x x var( x var(=1 x) n sd(x ) =) = n x ) = var(x ) sd( sd(x ) = var(x )
i.e. the mean ofx x ) 2 n=1 x i 2 the square of the mean n ( the squares iminus x) x = x2 var(sd(=) =i =1 ix ) var( n x ) = n var(x ) (sd) is the square root of the sd( The standard deviation variance: n n i =1 ln AB = ln A + ln B ; ln A = ln A ln B ; ln A n = n ln A B then multiply their denominators: e.g.
3 2 3 3 9 3 5 = 15 2 = 10 5 3 3 =16 15 5 4 31 7+ 11= 77 + = 5 4 20 20 20 3 = 5 = 15 Proportion and Percentage
To convert a fraction into a percentage, multiply by 100 5 and express the result as a 5 100 = 62.5% example is: as a percentage is percentage. An 8 8 5 5 as a percentage is 100 = 62.5% 8 8 Some common conversions are
1 1 1 3 = 10% = 25% = 50% = 75% 10 4 2 4 1 1 1 3 = 10% = 25% = 50% = Ratios are simply an alternative way 75% 10 4 2 4 of expressing fractions. Consider dividing 200 between two people in the ratio of 3:2. This means that for every 3 the first person gets, the second person gets 2. So the first gets 3/5 of the total (i.e. 120) and the second gets 2/5 (i.e. 80). sd(x ) = var(x )
Note that the standard deviation is measured in the same units as x. Dividing fractions 3 2 3 3 9
5 3 7 5 two fractions, invert the second and then 10 To divide112= 77 multiply: e.g. 3 2 3 = 5 3 5 3 9 = 2 10 The Greek Alphabet alpha beta gamma delta epsilon zeta theta iota kappa lambda nu xi omicron pi rho tau phi chi sigma upsilon Series (e.g. for discounting) 1 + x + x2 + x3 + x4 + ... 1 + x + x2 + x3 + ... + xk (where 0 < x < 1 ) = 1/(1x ) = (1x k+1 ) /(1x) mu eta psi omega Generally, to split a quantity in the ratio m : n, the quantity is divided into m/(m + n) and n/(m + n) of the total. ...
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 Spring '09
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